The highest value of electron-hole recombination time I know of is around 1 millisecond, in very high quality single-crystal silicon. This material has very little defect-based (SRH) recombination (because of the high quality), and very little radiative recombination (because it has an indirect gap), and very little Auger recombination (if the doping is not too large). 1 millisecond is extremely unusual, a triumph of engineering. Nanoseconds would be more usual. Sometimes in high-speed devices, the semiconductor is purposefully engineered to have unusually low lifetime.
Recombination tends to occur faster at higher temperature, although I don't think it's a vast difference in most cases. It depends on the dominant recombination mechanism. Certainly, phonon-mediated recombination would occur appreciably faster at high temperatures. Likewise, the effect of changing the band gap depends on what the dominant recombination mechanism is, including factors like whether the band gap is direct or indirect, and where the impurity levels sit relative to the band edges, and how the band gap change alters carrier densities (in the case of Auger recombination).
If you attach a semiconductor to leads, it will have Johnson–Nyquist noise related to its resistance and temperature, just like anything else. The fact that the electron thermal agitation may involve jumping from the valence to the conduction band does not make any special difference, except insofar as it affects the overall resistance.